Actual and potential infinity are not different "kinds of infinity". They have to do with different ways of dealing with infinity. When proving a theorem by natural induction -- say: Gauss' sum formula for finite arithmetic progressions-- you have been dealing with potential infinity: the formula is verified for each natural number. When using the set of all natural numbers in set theory, you are effectively using an infinite object. You are now dealing with actual infinity.

Cantor's infinite cardinal numbers can indeed be seen as degrees of infinity. Using expressions like "more infinite" for this make it sound ridiculous, but that is just playing with words. That the cardinality of the real number system (the continuum) is larger than the cardinality of the natural number system corresponds with a fundamental fact that one cannot run all reals in a counting process (Cantor's diagonal argument). It also demonstrates that a majority of reals are transcendental. Similarly, the power set of a set S outperforms the original set S by a famous argument of Cantor, which may have inspired the Russell paradox. This (so-called) paradox was just a warning that Cantor's paradise doesn't come for free; it can only be entered with a suitable axiom system for set theory.

I don't see the exact problem of this question. You can adhere to finitary mathematics if you wish, rejecting (actual) infinity. If not, you must face the fact that the concept of infinity comes with refinements which should not be approached with naive intuition. It's a technical thing.

This question has a lot to do with the philosophy of mathematics.

Endlessly Big or Endlessly Small, Unity being the Comparison.

Actual and potential infinity are not different "kinds of infinity". They have to do with different ways of dealing with infinity. When proving a theorem by natural induction -- say: Gauss' sum formula for finite arithmetic progressions-- you have been dealing with potential infinity: the formula is verified for each natural number. When using the set of all natural numbers in set theory, you are effectively using an infinite object. You are now dealing with actual infinity.

Cantor's infinite cardinal numbers can indeed be seen as degrees of infinity. Using expressions like "more infinite" for this make it sound ridiculous, but that is just playing with words. That the cardinality of the real number system (the continuum) is larger than the cardinality of the natural number system corresponds with a fundamental fact that one cannot run all reals in a counting process (Cantor's diagonal argument). It also demonstrates that a majority of reals are transcendental. Similarly, the power set of a set S outperforms the original set S by a famous argument of Cantor, which may have inspired the Russell paradox. This (so-called) paradox was just a warning that Cantor's paradise doesn't come for free; it can only be entered with a suitable axiom system for set theory.

I don't see the exact problem of this question. You can adhere to finitary mathematics if you wish, rejecting (actual) infinity. If not, you must face the fact that the concept of infinity comes with refinements which should not be approached with naive intuition. It's a technical thing.

Lexically "infinite" is a negation. Means NOT FINITE. You cannot say that something is more NOT FINITE as something else, and you cannot say that something is LESS FINITE than something else. Sets with three elements and with 10^27 elements are both finite. They have just different cardinalities. Also the sets N and R are both infinite. They have just different cardinalities. Cardinality and finity are two different criteria (criterion means here an equivalence class of sets). One Cardinality is the class of all sets with a given cardinality. Of course Finity is the union of all finite Cardinalities and Infinity is the union of all infinite Cardinalities. Like different Cardinalities, if compared, Finity and Infinity are disjoint classes of sets, and this is good so, because there is no set which is both finite and infinite. Also Cardinality 5 is disjoint of Cardinality 4, because there is no set which has in the same time 4 and 5 elements. This behaviors of Cardinality and (In)Finity are similar, and one tends to confuse them, as in our question. But they are different properties and must not be confused.

The question of actual vs. potential is much more a characteristic of our interaction with the concept of infinity, because this concept does not play all the time the same role in a phrase or argument. I must stress: the concept means every time the same thing, but can be used in different ways. Actual and potential is more about how do you use it. 

Dear friends,

According to Mr. Mohamed El Naschie, we can really have many different definitions with different natures for the concept of “infinite” in human science; but according to Mr. Tomas Kačerauskas and Mr. Subhash C. Kundu, “infinite”, “potential infinite”, “actual infinite”, “higher infinite”, “lower infinite”, “the ‘infinite’ of more infinite”, …are in fact not different "kinds of infinity". Such debates have been lasting since antiquity, that is OK and let it goes.

The practical problem is when facing and treating the “infinite items in infinite Harmonic Series” in present cluttered, unsystematic classical self-contradictory “infinite” theory system, we are unavoidable to meet following questions: why the Un--->0 Harmonic Series can be changed into an infinite series with items each bigger than any positive constants (such as 10000000000), what is infinite, what is infinitesimal, what is infinity? Cardinality, continuum hypothesis and non-standard analysis theories help nothing here.

This is the “2500-year-old huge black cloud of infinite related paradoxes over mathematics sky”.

Yours,

 Geng

Geng, you keep amazing me for seeing problems with the diverging Harmonic series "sum of 1/n for n=1 to infinity". Do you know that even "sum of 1/p for p prime" goes to infinity? That statement is stronger than the one about the Harmonic series.

And do you know that there is a conjecture of Erdos claiming that if the sum of the reciprocals of the members of a set S of natural numbers diverges, then S must contain arbitrarily long arithmetic progressions? And that this conjecture is connected with a truly famous result of Szemerédi (the Stone of Rosetta for mathematics, according to some)? And with a result that the set of prime numbers contains arbitrarily long arithmetic progressions?

All this is fine mathematics with currently open problems. It is a flourishing area of research, not a troublesome area full of so-called paradoxes, or, in your terminology, with a huge black cloud over it.

https://en.wikipedia.org/wiki/Erdos_conjecture_on_arithmetic_progressions

https://en.wikipedia.org/wiki/Szemeredi's_theorem

An 'aside' to the valuable posts above: Maybe we are locked up in our mathematical formalization and language when we try to define a concept. Quotations from 'human science' (as Geng put it) or philosophy or even literature enrich the mathematical meaning. One of my favorites from Shakespeare's Othello, Act I, Sc. III: '...my particular grief Is of so flood-gate and o'erbearing nature That it engluts and swallows other sorrows And it is still itself.' For me, this is clear definition of infinity accepted in mathematics almost 300 years later. Brabantio's 'private' grief is so great that adding a piece of 'public' grief would not change its (let us call it so) cardinality. A mathematical thought disguised in the form of a literary drama!

Making an intuitive idea precise is what mathematics does best, but it doesn't always eliminate philosophical controversy.  One confusion regarding the word "infinite" has to do with the difference between arithmetic and geometric intuitions.  For example, a cube or sphere in space is "infinite," in the sense of cardinality as a point set.  On the other hand, it is "finite," in the sense of geometric extent.  Sometimes in the popular science press you read that such-and-such observation implies that the physical universe is "finite but unbounded."  This is somewhat imprecise and misleading; but such language as "compact and without boundary" is far too technical for a newspaper article.  By the way, a similar problem arises with trying to make mathematical sense out of the intuitive notion of "computation." 

Dear Mr. Mihai Prunescu，

Thank you very much for your frank ideas.

In present science theory system, the concepts of “finite” and “infinite” have been defined merely through “quantities (endless, unlimited or not)” and we define (call) all those mathematical things with endless or unlimited quantities “infinite”.  Is this for sure?

But now there have been many different definitions for “infinite” just only because of the differences of “quantities”, such as “the quantity in ‘INFINITE A’ is more endless and more unlimited than the quantity in ‘INFINITE B’, so ‘INFINITE A’ is more infinite than ‘INFINITE B’” (the elements in Real Number Set is more endless and more unlimited than those in Natural Number Set, so Real Number Set is more infinite than Nature Number Set).

One may say, well we have to redefine “what endless or unlimited is”. Is this for sure, too?

You are right Mihai, the root is “actual vs. potential”.

thank you again Mr. Mihai Prunescu.

Yours,

Geng

Dear Mr. Marcel Van de Vel ，

Thank you very much for your opinions.

According to our studies in science history, there is a Zeon’s Paradox Family with many family members. Harmonic Series Paradox is the one serve as an example and one is enough.

The suspended infinite related Zeon’s Paradox is still there, that is why I say “2500-year-old huge black cloud of infinite related paradoxes over mathematics sky”.

Thank you again Mr. Marcel Van de Vel.

Yours,

Geng

Dear Mr. Paul Bankston and Mr. Paul Bankston ，

Your opinions open another window for us. Thank you!

It may be important for us to redefine “what endless or unlimited is” for the mathematical meaning.

Yours,

Geng

If one approaches the question from psychology and the history of mathematics, then there are different infinities. Adding 1 to the natural numbers can go on forever and therefore there is no last number, not even infinity. Similarly, successive subtractions from -1 can go on forever without there being a last negative number. Pairing the naturals with set of odd numbers producing yet another kind of psychological infinity and requires understanding a proof Then there are the infinite divisibles, where it is always possible to put another number between any two points on a line (no matter how close); infinite space; and so on. There is reason to think that people consider these different, as have philosophers, logicians and mathematicians, over the centuries.

Dear Ms. Rochel Gelman，Thank you very much for your opinions.

We call all those mathematical things with endless or unlimited quantities “infinite” such as the things in your post “Adding 1 to the natural numbers can go on forever and therefore there is no last number, not even infinity”.

It is mentioned many times in RG that people integrate “infinite” with psychology. It is ok that people can understand things in their own way; but in front of practical mathematical case, psychological understanding can not help, such as the strictly proven Harmonic Series Paradox:

The following proof (given by Oresme in about 1360), very elementary but important, can be found in many current higher mathematical books written in all kinds of languages.

1＋1/2 ＋1/3＋1/4＋．．．＋1/n ＋．．．                                   （1）

=１＋1/2 ＋（1/3＋1/4 ）＋（1/5＋1/6＋1/7＋1/8）＋．．．      （２）

>1+ 1/2 ＋( 1/4＋1/4 )+（1/8＋1/8＋1/8＋1/8）＋．．．           （3）

=1+ 1/2 + 1/2 + 1/2 + 1/2 + ．．．------>infinity                         （4）

Each operation in this proof is really unassailable within present science theory system. But, it is right with present modern limit theory and technology applied in this proof that we meet a “strict mathematical proven” modern version of Ancient Zeno’s Achilles--Turtle Race Paradox[1-3]: the “brackets-placing rule" decided by limit theory in this proof corresponds to Achilles in Zeno’s Paradox and the infinite items in Harmonic Series corresponds to those steps of the tortoise in the Paradox. So, not matter how fast Achilles can run and how long the distance Achilles has run in “Achilles--Turtle Race”, there would be infinite Turtle steps awaiting for Achilles to chase and endless distance for him to cover, so it is of cause impossible for Achilles to catch up with the Turtle; while in this acknowledged modern divergent proof of Harmonic Series, not matter how big the number will be gained by the “brackets-placing rule" (such as Un’ >10000000000) and how many items in Harmonic Series are consumed in the number getting process by the “brackets-placing rule", there will still be infinite Un--->0 items in Harmonic Series awaiting for the “brackets-placing rule" to produce infinite items each bigger than any positive constants, so people can really produce infinite items each bigger than 1/2, or 100, or 1000000, or 10000000000,… from Harmonic Series and change Harmonic Series into an infinite series with items each bigger than any positive constants (such as 10000000000), “strictly proving” that Harmonic Series is divergent. In so doing, the conclusion of “infinite numbers each bigger than any positive constants” can be produced from the Un--->0 items in Harmonic Series by brackets-placing rule and Harmonic Series is divergent" has been confirmed as a truth and a unimpeachable basic theory in our science (mathematics) while “the statement of Achilles will never catch up with the Turtle in the race” in Ancient Zeno’s Achilles--Turtle Race Paradox has been confirmed as a “strict mathematical proven” truth and a unimpeachable theorem------it would be Great Zeno’s Theorem but not Suspended Zeno’s Paradox! ?

Yours,

Geng

Dear Geng, nobody says "more endless" or "less endless". If so, I beg for a citation. Give us the references!

What they say is that there are infinite sets A and B such that there are injective applications i : A ---> B and surjective applications s: B ---> A but there don't exist surjective mappings s1: A ---> B or injective mappings i1: B ---> A. This means card(A) < card(B). The fact that we have already natural examples like A = N and B = R is wonderful. OK, some people may argue that R is only an imaginary construct, that the operation of completion applied to the geometric line is illusory. Might be, but there are still the infinite sequences of digits containing one point anywhere. So if N exists, then R exists as well, and has as well a bigger cardinality as N. There are also other people, as for example the late Edward Nelson in some of his works, who do not believe that even N does exist. They believe that all is finite, and even they believe that the Finity is bounded. 

What about you, dear Geng? Do you believe that infinite sets exist? Or even better, do you believe that N does exist? Because if you believe the existence of N, then you must accept following logical consequences the whole set theoretic architecture, and specially the fact that card (N) < card(R). 

And this must happen also if you only potentially believe the existence of N, but actually do not. Then you must potentially accept that card(N) < card(R). For actually checking the infinity of N we have of course not enough time, because our life is actually bounded.

Dear Mr. Mihai Prunescu，thank you.

1， I surely believe all the infinite sets being studied exist-------they are one form of “infinite carriers”.

2, the quantity of elements in Natural Number Set is infinite-------endless and unlimited, now the elements in Real Number Set has more in quantity than “the infinite” in Natural Number Set------anyone can say logically, less endless, less unlimited and less infinite just because of the fact that card (N) < card(R).

Unless we have to redefine “infinite”.

Yours sincerely,

Geng

Geng, please be clear about what is a paradox. Here is Webster's definition:

A tenet or proposition contrary to received opinion; an

assertion or sentiment seemingly contradictory, or opposed to

common sense; that which in appearance or terms is absurd,

but yet may be true in fact.

Loosely: a paradox is a statement that threatens common sense or common opinion. The strict mathematical usage is: a logical contradiction. One such contradiction is enough to corrupt the whole of mathematics, a disaster of unprecedented proportion. Since the axiomatization of set theory about a century ago, mathematics has not encountered such a "genuine" paradox.

Therefore, the mathematical usage of the term has gradually been downgraded to cover (more or less) what Webster's dictionary suggests. Such "paradoxes" are curiosities, jewels of ingenuity deserving admiration: the Banach-Tarski paradox, Peano's space-filling curve, many challenging counterexamples in (topological) dimension theory, etc.. All these assertions have an impeccable proof by all logical standards. 

Zenos "paradox" doesn't even belong to this list, as the argument is not based on logic but on naive common sense (technically, it is plainly wrong, or unjustified, depending on ones view). It does fall under Webster's definition, and it is good enough to serve as a paradox in philosophical circles.

Just as finiteness can be specified with natural numbers, infinity can be specified with cardinal numbers, too. Geng's remarkable attitude towards infinite cardinal numbers ("more, or less, endless" makes no sense) is a reminder that natural numbers really are benchmarks in the "fuzzy" area of finiteness in just the same way as (infinite) cardinal numbers are benchmarks in the infinity zone.

We are so used to the (decimal) notation of numbers that we no longer notice their technicality and the underlying theory. This technical trick helps to decide at once that 9800 is larger than 8999. If you would be given two heaps of things with these amounts, you would have a hard time figuring out which is the largest. Without good notation and theory, one may even be inclined to think that all large quantities are "the same" as some primitive cultures seemed to do.

Also, some "classical paradoxes" on infinity are "foreseeable" in the finite realm. Consider, for instance, the Galileo-type paradox that the even numbers are only half  of the natural numbers and yet their "quantity" is not different. If you would have a fortune of 10^1000 euros in material one-euro coins and a thief would steal half of this, the universe wouldn't live long enough for you to notice that you have been robbed. Similarly, applying Hilbert's trick (to accommodate a new guest in a full infinite hotel)  to a finite super-giant hotel, it may take a long time before experiencing that (the serial version of) the procedure gets cluttered.

https://en.wikipedia.org/wiki/Galileo's_paradox

https://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel

Dear Colleagues,

Good Day,

This is a different natures for the concept of “infinite”, please see

In non-standard models of the real number system, infinitesimals occur together with "infinite" numbers. Infinitesimals and infinite reals are connected by reciprocity: x vs 1/x. But George is right in that infinitesimals and infinite numbers belong to the language of ordered fields, while "infinity" in Geng's discussion belongs to the language of set theory and is associated with measuring set sizes. 

I can't resist sharing an experience with you. It shows my personal relation to infinity. I found a way of creating periodic numbers in the factorial number system. For example, e/3=0,1213425637849(10)5... This means that ALL the digits of this irrational number become known and familiar for me. Such findings give the delight (or just the illusion) of finding new unknown gates to the mysterious world of infinity.

Dear Colleagues, let’s see following 2 points?

1, In present science theory system, the concepts of “finite” and “infinite” have been defined merely through “quantities (endless, unlimited or not)”.

2, Infinite Set A is smaller than Infinite Set B (Cardinal A is smaller than Cardinal B), so Infinite Set A has less elements than Infinite Set B (to compare with Infinite Set B, Infinite Set A is not endless or unlimited and it proves that Infinite Set A is not infinite at all, it should not be called “infinite”). If we are forced to call them both “infinite”, we have to say Infinite Set B is more endless or unlimited than Infinite Set A------ Infinite Set B is more infinite than Infinite Set A (Infinite B is more Infinite than Infinite A).

Dear Miguel，

1, Infinite is not only for sets.

2, The adjective “proper" in “proper subset" is another mysterious, anything can be “proper" or “improper"; what is more, “transfinite” is in fact meaning “Infinite B is more endless or unlimited, more Infinite than Infinite A”. That is what I said “Mr. G. Kantor has tried to develop some different “infinite” theories specially (only) for set theory”-------- the “cardinal” related set theory: cardinal bijective theory, and the “non--cardinal” related set theory: Infinite Set B is more endless or unlimited than Infinite Set A (transfinite number, continuum hypothesis, uncountability proof of real number set, …).

3, If as you said “transferring common language to formal mathematics, something that is not always possible” than you actually mean “something should be developed to make the transferring possible in our science and mathematics”.

Best regards,

Geng

Hi,

Re:  "The adjective “proper" in “proper subset" is another mysterious, anything can be “proper" or “improper"; etc." 

You are doing it again. Proper/strict subsets, etc are not mysterious in the context of math. They are very well defined. Mathematics has formal (read technical) definitions of all concepts which usually do not reflect our everyday meaning. You could equally question many other areas of math. For example, in real life multiplication always increases the final amount and division decreases it, but not in math. Then there are negative numbers which do not exist in real life but exist in math. And so on.  Note that the dissonance with common language could also be found in physics and other sciences. Newton's laws is one example of this. Einsteins theories is another one. 

The point is that mathematicians create technical/formal definitions which often contradict common experiences reflected by our languages. Infinity and infinitesimals are just one of many examples of this technical approach. You cannot dispute their integrity in the context of their assumptions and definitions. You can, however, question assumptions, that is, accept, reject them or create new ones, and on this basis formalise your own ideas by developing a technical language which coherently describes them. So far you haven't done that. 

RE: "If as you said “transferring common language to formal mathematics, something that is not always possible” than you actually mean “something should be developed to make the transferring possible in our science”.

True, but this is an area of cognition and education, and not mathematics alone. 

Regards, UM

Dear Mr. Untangling Math，

Set N is a subset of Set R, why it is an “improper subset" so that Set N can not be put into a bijective relationship with Set R?

That is what I said “Mr. G. Kantor has tried to develop some different “infinite” theories specially (only) for set theory”-------- the “cardinal” related set theory: cardinal bijective theory, and the “non--cardinal” related set theory: Infinite Set B is more endless or unlimited than Infinite Set A (transfinite number, continuum hypothesis, uncountability proof of real number set, …).

Best regards,

Geng

Hi, 

It appears to me that you suggest that there are two types of infinity one related to cardinals and the other, as you describe, to "non-cardinals". This could be true in the case of human intuition, but not in mathematics.

As far I understand, there is only one kind of infinity as defined by Kantor theory in which the cardinal and ordinal aspects are always considered together. Separating them leads to logical problems. The separation of these two aspects is a common problem for students, which is what Miguel commented about. 

Regards, UM

Dear Mr. Untangling Math，

 Would you please tell me your frank view points to following statement: Set N is a subset of Set R, why it is an “improper subset" so that Set N can not be put into a bijective relationship with Set R?

Best regards,

Geng

Dear Mr. Untangling Math，

 Would you please tell me your frank view points to following statement: Set N is a subset of Set R, why it is an “improper subset" so that Set N can not be put into a bijective relationship with Set R?

Best regards,

Geng

Dear Miguel，

1, would you please tell me your frank view points why Set N is not a proper subset of Set R?

2, four mistakes have been discovered in that Cantor’s diagonal proof.

Yours,

Geng

Dear Miguel，

In fact, so many “countable subsets of R” are proper subset of R, but they all can’t be put into a bijective relationship with R------less endless or unlimited, less Infinite?!

Yours,

Geng

Dear Miguel，you are fight to the point.

 Why all numerable subsets of R cannot be put into a bijective relationship with R? There is only one answer: they have different quantity ------ because of less endless, less unlimited and less infinite?!

Yours,

Geng

Any numerable(countable) set is in bijective relationship with the set of natural numbers N. Since "bijection" is an equivalence relation, one need only consider whether or not there is a bijection between N and R. However, it is sufficient to consider a subset of R since we are showing it not to be so!

Using the convention that each real number corresponds to a unique non-terminating decimal representation is what allows for the "diagonalization argument". Once the real numbers between 0 and 1 are "listed", there are multiple ways of defining an existing decimal representation which does not exist in the list. Thus, a subset of R is uncountable. Therefore, R is uncountable.

Dear Mr. Geoff Diestel，

Yes, "bijection is an equivalence relation "!

Now a very clear and generally accepted situation is: N is sour to have fewer elements than R in our present science theory system, thus it is logical for anyone to say “N is less endless, less unlimited and less infinite than R”. So, that is why we may have some other expressions (hypothesis) logically: less infinite, less less infinite, less less less infinite,…

What is infinite?!

Yours,

Geng

Hi Geng and others,

Once again, I would suggest the article by Dong-Joong Kim, et al titled: "How does language impact the learning of mathematics? Comparison of English and Korean speaking university students’ discourses on infinity" (https://www.researchgate.net/profile/Anna_Sfard/publications). The article suggests that

“In Korean, unlike in English, there is a disconnection between colloquial and mathematical discourses on infinity, in that the mathematical word for infinity is not a formalized version of a colloquial word but a novel sound, inspired by a Chinese term for infinity.”

Further more,

“In general, whereas no group could pride itself on a well-developed mathematical discourse on infinity, the mathematical discourse of the English speakers, just like their colloquial discourse, was predominantly processual, whereas the Korean speaking students’ talk on infinity was more structural and, in an admittedly superficial way, closer to the formal mathematical discourse.”

Cantor defined infinity seem to be, correct me if I am wrong, at least in part processual. For example, injection/bijection rule only works as long as new numbers could be generated. It is counterintuitive because it suggests that that all subsets of countable numbers are equal in “size”.

In the context of never ending processes, the notion of size has no meaning. One can only say that a subset is in bijective relationship, or not. And if it is, then the two sets could be described as equivalent. This technical definition of equivalency has nothing to do with our everyday experience of size or greater/ equal/ smaller relationship (e.g. fewer elements).

One can accept this approach or rejected it, and create another form of mathematics based on different assumptions and procedures. But you cannot say that definitions and procedures adopted by Cantor are somehow wrong, the same way as you cannot say that the rules of chess are wrong. They are what they are.

Geng, I would appreciate it if you could address in some detail the issues outlined in the article by Dong-Joong Kim. There are some differences in our approaches to infinity and we need to understand why. 

Best, UM

Dear Mr. Untangling Math，you are very frank, friendly and kind, thank you very much.

1, after reading the article of Mr. Dong-Joong Kim, I understand the differences between “English and Korean speaking university students on infinite” three authors talk about are in fact “differences between actual infinite and potential infinite”. So far as I know, such “debates between actual infinite and potential infinite” have been in our science by all kinds of languages (not only by “English and Korean speaking university students”) for at least 2500 years since Zeno’s time with his suspended paradoxes. Such “debates between actual infinite and potential infinite” will last forever in present chaos infinite theory frame not only by different language speakers but just by researchers, nothing to do with languages.

2, in front of practical mathematical case, we need clear understanding by all researchers (nothing to do with languages), such as the strictly mathematical proven Harmonic Series Paradox:

The following proof can be found in many current higher mathematical books written in all kinds of languages.

1＋1/2 ＋1/3＋1/4＋．．．＋1/n ＋．．．                                        （1）

=１＋1/2 ＋（1/3＋1/4 ）＋（1/5＋1/6＋1/7＋1/8）＋．．．      （２）

>1+ 1/2 ＋( 1/4＋1/4 )+（1/8＋1/8＋1/8＋1/8）＋．．．              （3）

=1+ 1/2 + 1/2 + 1/2 + 1/2 + ．．．------>infinity                                （4）

Each operation in this proof is really unassailable within present science theory system. But, it is right with present modern limit theory and technology applied in this proof that we meet a “strict mathematical proven” modern version of Ancient Zeno’s Achilles--Turtle Race Paradox[1-3]: the “brackets-placing rule" decided by limit theory in this proof corresponds to Achilles in Zeno’s Paradox and the infinite items in Harmonic Series corresponds to those steps of the tortoise in the Paradox. So, not matter how fast Achilles can run and how long the distance Achilles has run in “Achilles--Turtle Race”, there would be infinite Turtle steps awaiting for Achilles to chase and endless distance for him to cover, so it is of cause impossible for Achilles to catch up with the Turtle; while in this acknowledged modern divergent proof of Harmonic Series, not matter how big the number will be gained by the “brackets-placing rule" (such as Un’ >10000000000) and how many items in Harmonic Series are consumed in the number getting process by the “brackets-placing rule", there will still be infinite Un--->0 items in Harmonic Series awaiting for the “brackets-placing rule" to produce infinite items each bigger than any positive constants, so people can really produce infinite items each bigger than 1/2, or 100, or 1000000, or 10000000000,… from Harmonic Series and change Harmonic Series into an infinite series with items each bigger than any positive constants (such as 10000000000), “strictly proving” that Harmonic Series is divergent. In so doing, the conclusion of “infinite numbers each bigger than any positive constants” can be produced from the Un--->0 items in Harmonic Series by brackets-placing rule and Harmonic Series is divergent" has been confirmed as a truth and a unimpeachable basic theory in our science (mathematics) while “the statement of Achilles will never catch up with the Turtle in the race” in Ancient Zeno’s Achilles--Turtle Race Paradox has been confirmed as a “strict mathematical proven” truth and a unimpeachable theorem------it would be Great Zeno’s Theorem but not Suspended Zeno’s Paradox! ?

Dear Mr. Untangling Math, could you please give your frank views on above two points?

Thank you again!

Best regards,

Geng

 UM, a minor correction about having only "one kind of infinity". As far as set theory is concerned there is indeed no rivaling concept. But the qualifications "infinite element" as well as "infinitesimal" are common concepts in the area of non-Archimedian fields, with non-standard real number systems as prominent models.

Dear Mr. Marcel Van de Vel，

As is known that “non-standard real number system” is equivalent with “standard real number system” within present science theory frame, it is in fact a special infinite number theory purposely designed for analysis only (the exactly same thing happens in present set theory).

So, in front of Zeno’s Paradox Family, “non-standard real number system” is exactly like the present “standard one” can do nothing at all.

An infinite theory system should be composed by three parts:

1, infinite concept system.

2, infinite number system (infinite things including sets)

3, treating theories and techniques for “infinite things”

Our science history has proved that It is very easy to make many models for the second and the third in present science theory system but it is very difficult in present science theory system to own the first------building a scientific infinite concept system free from paradoxes. Because the suspended paradoxes have proved that this work relates to a huge working field and there are many “unscientific things” needed be improved in philosophy, mathematics, cognizing theory, …

Best regards,

Geng

Dear Geng, a (maybe naive) remark to your statement: 'It is very difficult to build a scientific infinite concept system without paradoxes.' Is it possible that the paradoxes arise because 'infinite' is a negative definition that does not state what the defined concept is, but only what it is not? (Clumsy English, but I could not find a better wording even in my native Hungarian.) Is it possible that you can find similar paradoxes with any 'non-definitions' such as 'incomplete', 'colorless', 'meaningless', or 'indefinite' as well?

Dear Mr. István Lénárt，

Thank you very much for your friendly and frank points of view. I beg your pardon, English is my second language, so very often I may not be able to express my idea clearly but I will learn more and try my best. I have made a correction on that post, but I am still not sure enough whether it is ok now.

What I really mean in that statement is: it is very difficult in the frame of present science theory system to build a scientific infinite concept system free from paradoxes. Because the suspended paradoxes have proved that this work relates to a huge working field and there are some “unscientific things” needed be improved in philosophy, mathematics, cognizing theory, …

Thank you again!

Best regards,

Geng

Hi Geng,

Regarding paradoxes, in my opinion, there are two types of paradoxes: mathematical ones and the rest.

A typical example of mathematical paradoxes is Russel paradox which exposed certain inconsistencies of one particular set theory. Such paradoxes expose shortcomings of definitions and axioms of a given theory. They are more of a ‘proof’ than paradox as such.

Zeno paradoxes are examples of non-mathematical arguments. There are no definitions or axioms to disprove. They are just words games appealing to our sense of amusement. They demonstrate that problems need to be defined in an unambiguous way to avoid paradoxical situations. 

Dear Mr. Untangling Math,

According to my studies, Zeno’s Achilles--Turtle Race Paradox and its modern version-------above Harmonic Series Paradox I presented 5 days ago in this thread is a very typical “mathematical ones” because the Harmonic Series Paradox is a mathematical problem. It is for sure that in present mathematics theory system, any infinite related paradox has a lot to do with philosophy, mathematics, cognizing theory, …

Best regards,

 Geng

 Hi, 

"Achilles and the Tortoise":  In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead. 

Here is what Wolfram MathWorld has to say on the topic”:

“Dichotomy paradox: Before an object can travel a given distance d, it must travel a 1/2 distance. In order to travel d/2, it must travel 1/4, etc. Since this sequence goes on forever, it therefore appears that the distance d  cannot be traveled. The resolution of the paradox awaited calculus and the proof that infinite geometric series  can converge, so that the infinite number of "half-steps" needed is balanced by the increasingly short amount of time needed to traverse the distances.”

As you can see, one can interpret the text as a mathematical problem and show that the statement is not true. Alternatively, you can concentrate on the sequence alone by ignoring the contex. Only when you combine the two aspects together, the statement sounds like a paradox.

In summary, the "Achilles and the Tortoise" story demonstrates that most, if not all, people find conceptualising infinity very difficult.

Dear Mr. Untangling Math, thank you and I am very happy to have discussions with you.

How do you think of the mathematical Harmonic Series Paradox I presented 7 days ago in this thread? It is nothing to do with conceptualising infinity, just mathematical operations.

Yours,

Geng

Since the concept of infinite was born in our science, the question of “What is potential infinite and actual infinite?” has been analyzed, discussed and debated and this situation is sure to be “endless” in present classical science theory frame--------our science history strongly proved!

What have happened in set theory and analysis also strongly proved that infinities and infinitesimals are both visible and touchable needed in our science on both “theoretical and practical or applied” level. Meanwhile, the suspended infinite related paradoxes families since Zeno’s time more than 2500 years ago have keeping proving that something must be wrong in the foundation of present classical science theory frame!

It is a dilemma in front of us human: to be or not to be staying in the defected foundation of present classical infinite related science theory frame.

Thank you my dear Mr. downvoter, would you be so kind as to tell me your frank point of view what mistakes in the above post and how to correct the mistakes.

Beg you and thank you again.

Yours,

Geng

Hi Geng, 

Regarding my post about Zeno's paradoxes:

Any high school student familiar with Cartesian coordinates can easily demonstrate that mathematical/physical problem posed by "Achilles and the Tortoise" story is ill conceived. It misuses the concept of movement which by definition involves both distance and time. By formulating the problem only in terms of distance, Zeno is able to create a paradoxical situation in which Achilles will never overtake the tortoise.

Was Zeno ignorant or careless?  I do not think so. To see this, one has to look at Zeno’s paradoxes in the context of his times. Zeno of Elea was a student of Parmenides who, in the argument with Heraclitus, used logic to disprove the possibility of motion. You can find the same preoccupation with motion and logic in all Zeno's paradoxes. The selective use of logic promoted by Zeno became later known as sophism. From this perspective, Zeno's paradoxes do not prove anything.  

Regards, UM

Hi Geng, 

Regarding: “Thank you my dear Mr. downvoter, would you be so kind as to tell me your frank point of view what mistakes in the above post and how to correct the mistakes.”

I did not vote you down, but I can imagine that others might be as frustrated as I am about little progress of our discussion about infinity/infinitesimals. Perhaps the most disappointing issue for me is that first you ask others to express their views and then you seem to ignore those views. Our discussion would be so much more exciting if only you could show some interest in those ideas, e.g. considered them.

Regards,UM

Thank you dear Mr. Untangling Math.

1, in fact, we can see some people believe that Zeno’s paradoxes are not “paradox” and these “Zeno’s paradoxes” didn’t prove anything or they believe that Zeno’s paradoxes have been solved. While some other people have totally opposite point of view. According to my studies, like “potential infinite and actual infinite”, such disagreements is sure to be “endless” in present classical science theory system--------our science history strongly proved! We may not involve into such disagreements debates dealing with time, space or movement before the fundamental defects in present classical science theory system are solved. The strict mathematical proved Harmonic Series Paradox has nothing to do with time, space or movement but has exactly the same trouble of "Achilles and the Tortoise Race". “The strict mathematical proved Harmonic Series Paradox” has really proved something.

2, as a second language, my English is poor and I sometimes have to ask for other people’s help; but I have tried my best to answer most questions in this thread and always feel very thankful for the researchers who have participated in this discussion. Now I understand I am culpable of punishment for somewhat I may ignore some views, maybe my RG scores has been going down and down for exactly the same reason.  Ok, forget so many “maybes”, that is the life. I will keep trying harder.

Thank you again.

Yours, Geng

To Geng Ouyang, Marcel Van de Vel, Miguel Ángel Montes and other adherents of the Cantor’s ideas.

Any mathematical concept must be based on mathematical praintuition (Brouwer's term for basic intuitive foundations of mathematics) which is immanent for all people (otherwise, mathematics cannot exist as common science). This mathematical praintuition gives the concept of the infinite series of integers to everyone. Therefore, this concept is accepted by all mathematicians and it was formalized (axiomatized) by Peano. This is the concept of potential infinity and any mathematical point of view must be consistent with this concept.

But Cantor introduced in mathematics the new (non-mathematical but religious) concept of actual infinity which is in a contradiction with the mathematical concept of infinity, i.e. potential infinity (since Cantor supposed that endless set can be bounded). Naturally, the contradictions (which are often called antinomies) of his set theory arose.

Thus, the actual infinity does not exist. There is the only kind of infinity: potential infinity. And the only kind of an infinite set: countable set. There is no procedure to define (to create) a so-called uncountable set. The only procedure which let Cantor create (as he thought) another infinite (not potential infinite) set, was the procedure of the creating of the power-set of infinite set. But he (and many other people) did not understand that the power-set of infinite set does not exist (for the same reason - this assumption leads to a contradiction). It is a very common error: the transference of the property of finite sets to an infinite set. And this is an error of any known system of axioms for the set theory: the axiom of a power-set is not valid for an infinite set, since there are no intuitive mathematical foundations for this axiom for an infinite set and this is the point where actual infinity appears. Thus, there is no method to create an uncountable set. And it is naturally and understandably: no one can create something which cannot exist. And, of course, there are no infinite numbers (nor cardinal neither ordinal).

Dear Mr. Vadim Khodorovskiy

I think in our present mathematical theory frame, people have to accept following two things:

1, infinite as a kind of endless, limitless, untouchable, invisible but imaginable things and you are unable to have any mathematical calculations on “infinite things”; so we have “potential infinite” and you can say about it, talk about it--------this concept is really there in our science;

2, infinite as a kind of un-endless, un-limitless, touchable, visible things and you are able to have all kinds of mathematical calculations on “infinite things”; so we have “actual infinite” for analyses and set theory-------this concept is really there in our science;

But the real situation is: in our present mathematical theory frame, these two infinite concepts are antinomies and many suspended paradoxes since Zeno’s time 2500 years ago were created to disclose those serious defects in the infinite related foundation in our present mathematical theory frame.

That is why we need A Revolution in the Infinite Related Foundation of Mathematics to dispel “the huge black cloud of infinite related paradoxes over mathematics sky”.

Yours,

Geng

Dear Geng,

You are right, when state that in modern mathematics there are two concepts of infinity, but, I should like to assert, that only one of them (potential infinity) is acceptable, whereas another one (actual infinity) is erroneous, since it is self-contradictory (inasmuch as it implies that infinite sequence of integers can be bounded and endless process can be ended). And all paradoxes of the set theory are the consequences of the self-contradictory idea of actual infinity.

Nevertheless, the Zeno’s paradoxes do not relate to actual infinity, since Zeno operated only with potential infinity (as far as I know). Therefore, his paradoxes are not the real ones, but they are seeming. E.g., "Achilles and the Tortoise" shows that Achilles will never overtake the tortoise, but really, the whole consideration is limited by finite time (let us note it by T), and hence, the paradox shows that Achilles will not overtake the tortoise before time T. As one can see, it is not a paradox.

Thus, regarding your last statement, I should like to say that it is very easy to make "A Revolution in the Infinite Related Foundation of Mathematics": the concept of actual infinity is to be rejected.

Hi Vadim,

Regarding : “the concept of actual infinity is to be rejected.”

We all use the concept you just excluded every day. Without it we would not be able to formulate our thoughts. For example, we can say "walking forever" or just "walking" could be very tiring/ exhausting/ difficult/ impossible/ etc. This construction is known as a continuous tense, which (among other things) could be used to talk about continuous actions as if they were completed, especially when combined with such words as consistently, constantly, ever, invariably, regularly, repeatedly, perpetually, eternally, etc. There are other English tenses which can be used in a similar way.

Consider such examples as “I am sleeping/ standing/ complaining/ existing/ etc” – one could sleep/ stand/ complain/ exist/ etc forever – and yet we can talk about those actions as if they were completed wholes.

Why can’t we do the same in math?

I think you are confusing mathematics with actual (sic!) reality.  

To Untangling Math

Although the continuous tense maybe used in the way you noted, but there is no connection between it and infinity (by the way, with "constantly", "ever", "regularly", etc. one should use simple tense, not continuous).

You, probably, confuse the literal sense of words and phrases with their true sense in a common speech. When people (in usual life, not in mathematics) say "infinitely many" they mean so huge amount, that it is too difficult to calculate or measure it, but this amount is finite. When people (and you in your post) say "forever" or "eternity" they mean a very long period of time, but this period is finite. Moreover, "infinity" is a pure mathematical concept which does not relate to the real world. Indeed, "infinity" means that we KNOW that some process CANNOT be ended. Such knowledge is possible only in mathematics, since it relates to our inner ideal world. The essential foundations of the real world are fundamentally unknown for us, and therefore, we cannot KNOW that something in this world CANNOT be ended. That is why infinity maybe only in mathematics, and this is, of course, potential infinity. (I should note that I talk about field of rational thinking, I do not concern any irrational fields of human activities such as religion).

Hi Vadim, 

Regarding: "When people (in usual life, not in mathematics) say "infinitely many" they mean so huge amount, that it is too difficult to calculate or measure it, but this amount is finite."

You did it again. You spoken about "infinitely many" or "infinity" as an object. 

Re: "forever"

Forever literally means for ever without end; everlastingly; eternally. It is often used metaphorically, that is, the way you described it in your  message. But literally, forever means a process without an end.

Re: "Moreover, "infinity" is a pure mathematical concept which does not relate to the real world." 

I agree with this statement. I see mathematical infinity as a purely technical concepts defined by its mathematical properties. Cantors approach to infinities is very much technical.  But even mathematical concepts require cognition, just like speaking or understanding. The mechanism of treating endless acts or processes as objects is called "encapsulation" which  was studied in depth in the educational context by Dubinsky and others (see e.g.        http://www.math.kent.edu/~edd/HistPart2RevSubmit1.pdf). 

I find the term "potential infinity" very confusing. What does it really mean - a never ending process which can potentially end? This is illogical. But treating endless processes as if they were objects is not. We do this everyday.  

Potential infinity actually is a philosophical term (introduced by Aristoteles). It can be used when talking about mathematics, but its is not itself a mathematical term. Consider, for instance, the true statement "for each prime number there is a larger prime number". This is a very clean way of saying that there are infinitely many prime numbers: given the prime 2, I find the larger prime 3; given 3, I find a larger prime 5, etc. This is an endless process which is elegantly avoided by the above "clean" statement (which has an admirable proof).

The statement "for each pair (p,q) of prime numbers with q-p=2 (twin primes) there is a pair of larger prime numbers which are twins" is not (yet) known to be true. Potential infinity is the hard issue here: computers may manage to find ever larger pairs of twin primes, but (as one cannot go on like this forever) these efforts don't prove the statement.

In present classical “multi--infinite--definition” related mathematics, when facing “infinite something”, one at lest have two troubles:

1, “potential infinite” or “actual infinite” or “super infinite” or “sub infinite” or “secondary infinite” or…?

2, how much “endless, limitless and infinite” it would be?

(a),When paying equal attention on “potential infinite” and “actual infinite”, we meet the uncompromised logical contradiction and the typical case is the “strict mathematical proven” ancient Zeno’s Paradoxes and modern Paradox of Harmonic Series.

(b), When paying more attention on “potential infinite” and neglect “actual infinite”, we meet all the ideas and theories of “all infinite things are the same endless without any inborn and numerical natures” and the typical cases are all the ideas and theories of the countable infinite sets.

(c), When paying more attention on “actual infinite” and neglect “potential infinite”, we meet all the ideas and theories of “all infinite things are not the same endless, they have inborn and numerical natures” and the typical cases are all the ideas and theories of uncountable infinite sets: more infinite, more more infinite, more more more infinite,…; higher infinite, higher higher infinite, higher higher higher infinite,…; bigger infinite, bigger bigger infinite, bigger bigger bigger infinite,…; super infinite, super super infinite, super super super infinite,…; Continuum Hypothesis,… 

Can we really have many different definitions with different natures for the concept of “infinite” in human science?

Can we name any infinite things “potential infinite” or “actual infinite” or more infinite or supper infinite or… as we like?

Can we pay attention on “potential infinite” and neglect “actual infinite” or vice-versa as we like?

…

That is why we need A Revolution in the Infinite Related Foundation of Mathematics to dispel “the huge black cloud of infinite related paradoxes over mathematics sky”.

To Untangling Math

 Re: "The mechanism of treating endless acts or processes as objects is called "encapsulation" which was studied in depth in the educational context by Dubinsky and others (see e.g. http://www.math.kent.edu/~edd/HistPart2RevSubmit1.pdf)".

I read this paper and its Part 1 and must say that there is nothing new in them, including errors. It turned out, that so-called "encapsulation", when it is applied to infinite processes, (with one exception) or is the well-known concept of limit (like in the explanation of the Achilles and the tortoise paradox) either is erroneous (like in the explanation of the tennis balls paradox (section 2.2.2, Part 1)). In the first case we can speak about the result (or resulting object) of an infinite process, since it has a limit (the concept of a limit does not concern actual infinity).

In the second case there is no limit and, hence, there is no resulting object. Thus, the question "what is the resulting object?" is meaningless. And there is no paradox here. But if one accepts the concept of actual infinity then the second paradox is the real one. And it is impossible to explain it by speaking or writing the "magic" word "encapsulation". The declared fact, that the sequence of sets, where each next set has one more ball, has the empty set as a resulting object is a real paradox. And this paradox illustrates or even proves the fact that actual infinity does not exist.

I mentioned one exception above, and this exception is the "encapsulation" (irrelevant term in this case) which yields the set of natural numbers N = {1; 2; 3; …} (section 3.1, Part 2). The natural series is the basic concept of an infinite set (and it is the only type of an infinite set) which mathematical praintuition gives us (in axiomatic set theories it is postulated by axiom of infinity). The expression N = {1; 2; 3; …} is merely convenient notation for this set. We can consider this set as an object, but the existence of this object is potential. This fact is expressed by "…" and means that we cannot attain its end since it does not exist. Therefore, for this set there are some restrictions for considerations and operations in comparison with finite sets. E.g., we cannot consider "all members" of it as a finished object or construct the power-set of it.

Hi Vadim,  

Many thanks for the answer. I have to commend you for reading the article the link of which I provided in my last message. This might seem as an obvious thing to do, yet it is not widely practiced on this and many other RG forums. After reading your answer, I can say that I have a somewhat better understanding of your thinking. I will tackle the statement in the last paragraph of your argument first as it seems to highlight our differences. 

Re: “The expression N = {1; 2; 3; …} is merely convenient notation for this set. We can consider this set as an object, but the existence of this object is potential. This fact is expressed by "…" and means that we cannot attain its end since it does not exist.”

My feeling is that you are preoccupied with the issue of existence. I am not sure, though, what kind of existence you have in mind. Based mainly on the articles published on your RG site, my guess is that you interpret the concept of existence in purely physical terms. This is perhaps where our approaches depart as I look at the issue of infinity mainly from a cognitive and mathematical point of view. The statement (you made and I quoted above) is perfectly logical in the context of your perspective and its assumptions about what is real. My approach is more liberal, though, mainly because it does not relate to reality but to its perception, that is, to the ways in which humans think.

The fact is that it does not matter whether something is physically possible - we can still talk about it with little or no problems. We all know that “unicorns” do not exist and yet we can like or despise them, debate about them, multiply or divide them, and so on. We can even talk about infinitely large sets of all unicorns.  Note that we can also manipulate variables (e.g. 2∙x) even before we know what they are; or tentative objects such as (a + b) even though the act of addition is not completed; and so on. This ability to talk about unknown or non-existing entities is not restricted to unicorns, unknown variables, uncompleted or never ending processes – it is the most basic feature of human cognition.

Re: “In the first case we can speak about the result (or resulting object) of an infinite process, since it has a limit (the concept of a limit does not concern actual infinity).In the second case there is no limit and, hence, there is no resulting object. Thus, the question "what is the resulting object?" is meaningless.”

Obviously we do not share the same understanding of the term “object”. For me objects are things I can “handle” either physically and/or mentally. The expression 2 + 3a – 4x2 is an object because I can multiply it as a whole even though the variable “x” and factor “a” are unknown, and operations of addition and subtraction are not completed. In the same way we can treat infinity as an object. Note that one can turn a process into an object by simply giving it a name. The expression N = {1; 2; 3; …} signals that we treat {1; 2; 3; …} as a single entity we call “N” so we can carry some mathematical operations on it.

You might say that my interpretation is unscientific because everything is permitted. This is not the case though. The approach is extremely restricted by the limited range of human expressions. For example, mathematical expressions are highly structured. There is no variation in their interpretation possible. In my research I look at expressions humans produce to make an inference about mental processes which led to the production of those expressions. I take mathematical expressions as they come. I do not dispute their correctness. Mathematics is what it is in the context of its definitions and assumptions. I just try to understand its meaning. That is all.  

I like to stop here to keep the message short. 

UM

(quoting from Vadim)

only one of them (potential infinity) is acceptable, whereas another one (actual infinity) is erroneous, since it is self-contradictory (inasmuch as it implies that infinite sequence of integers can be bounded and endless process can be ended).

Comment: Forming the set of all natural numbers is not an act of completing an infinite process. It could better be compared with a quite common act  (performed on a daily basis even in no-nonsense companies/enterprises): forming maps with a well defined subject, which are meant to hold all present and future documents falling under the subject (e.g., an insurance company may have maps labeled "car accidents", "fires", etc. You can operate on such maps (move the "car accidents" to a different shelf, subdivide the map "fires" into "home fires" and "company fires", etc. There is no process completion. The most frequent process is a process of recognizing that a subject x has or has not to be classified under label y. You can even classify labels under super-labels (shelves, for instance).

(quoting from Vadim)

And all paradoxes of the set theory are the consequences of the self-contradictory idea of actual infinity.

Comment: The most famous contradiction in set theory is the one of Russell. Here is the essence of it (where P(y,x) denotes any predicate about x and y):

(*)      (exists x) (all y)  (not P(y,y) P(y,x))

Suggested interpretation: There is a set x containing exactly all y that are not member of themselves. Or the barber paradox: P(y,x)="y is being shaved by x". Or any interpretation of P(y,x) you can think of. It always yields a nonsensical result. Note the absence of any reference to "infinity"; in fact, the barber version can be seen as a finite case.

Statement (*) is what formal logic calls a contradiction, just like "p and not p" (for any statement p). It is only hiding a bit deeper. The Russel's paradox occurred because it was thoughtlessly assumed that anything you could "collect by a mental act"  delivers a set. Wrong assumption: existence of sets has to be regulated by axioms.

Potential and actual infinity are philosophical terms that can be used in discussing how mathematics deals with infinity. Infinity is seen as a possible source of paradoxes because  (actual) infinity seems to have no counterpart in reality. But Peano arithmetic, considered to be a "true" representation of "the arithmetic reality", suffers from the same threat.

To Untangling Math

Re: "my guess is that you interpret the concept of existence in purely physical terms".

Of course (and fortunately), your guess is wrong. All our discussions relate to mathematics and therefore the physical existence does not matter. Moreover, mathematical objects cannot exist in physical sense, since they are the ideal objects of our inner world (e.g., integer or real numbers).

I mean traditional mathematical sense: "to exist" means "to be free from contradiction" (Poincare). We can produce any construction (concept) but it exists only if it is not contradictory. The concept of the actual infinity is contradictory (inasmuch as it implies that endless process can be ended). Thus, the actual infinity does not exist.

Re: "Re: “In the first case we can speak about the result (or resulting object) of an infinite process, since it has a limit … In the second case there is no limit and, hence, there is no resulting object. Thus, the question "what is the resulting object?" is meaningless.”

Obviously we do not share the same understanding of the term “object”."

 In this case my use of this term is generally accepted: if the sequence converges then it has a limit (this limit is an object); if the sequence diverges then it has no limit, i.e. there is no (resulting) object.

 Re: "Note that one can turn a process into an object by simply giving it a name."

Yes, you are right, but only if one defined this process before and if this definition is free from contradiction.

Re: "Mathematics is what it is in the context of its definitions and assumptions. I just try to understand its meaning."

Me too. And if I find out that some definition is contradictory then I conclude that the definition is meaningless and does not define any object.

To Marcel Van de Vel

Re: "Forming the set of all natural numbers is not an act of completing an infinite process."

Yes, of course. In this case we deal with potential infinity. The completing of an infinite process, in the most explicit and contradictory form, was introduced by Cantor (by class (I) he denoted the series of the natural numbers ν): "… although it is contradictory to say about the largest number of the class (I), there is no absurd to imagine some new number – denote it by ω – which must be the expression of the fact that whole set (I) is given for us according to its law in its natural sequence… One can even imagine the new-created number ω as the limit to which the numbers ν tend to, if one understand by this only that the number ω must be the first integer which comes behind all numbers v, i.e. which could be called larger than any number ν."

There may be different guesses, why Cantor saw no absurd in this idea, but one can see, as a result of unbiased thinking, that the idea is absurd. And whole set theory originate from this nonsensical idea.

Re: "The most famous contradiction in set theory is the one of Russell… Note the absense of the word "infinite"."

The absence of the word "infinite" does not mean the absence of infinity. The postulated set in the paradox is infinite. Unfortunately, this set does not exist as well as the Set of all sets. Only the absurd idea of actual infinity lead to the contradictory inference of the existence of such sets.

Hi Vadim,

Re: “Moreover, mathematical objects cannot exist in physical sense, since they are the ideal objects of our inner world (e.g., integer or real numbers).”

I am trying to anchor “our inner world” in cognition or, more precisely, in the way human sensory experiences are organised. How would you describe your interpretation of “our inner world”? To say that mathematics is this inner world is not satisfactory, at least to me.

Re: “I mean traditional mathematical sense: "to exist" means "to be free from contradiction" (Poincare). We can produce any construction (concept) but it exists only if it is not contradictory. The concept of the actual infinity is contradictory (inasmuch as it implies that endless process can be ended). Thus, the actual infinity does not exist.”

I am not a mathematician, but I am sure that Marcel Van de Vel would agree with me that there is no contradiction in this case. The problem is, as I see it, that you allow your intuition of physical experiences of never ending processes to interfere with the mathematical concept of infinity. To avoid this interference we should avoid any reference to “potential and actual infinity” in our discussion and in their place use mathematical symbols or other unrelated words. Moreover, the contradictions you refer to can only occur within the systems mathematician create, that is, in relation to definitions and axioms of that system. Mathematicians can define their objects any way they want to - some of those systems may be more useful than others, but they all can be internally consistent even if they are not intuitively obvious.

Re: “Obviously we do not share the same understanding of the term “object”." In this case my use of this term is generally accepted: if the sequence converges then it has a limit (this limit is an object); if the sequence diverges then it has no limit, i.e. there is no (resulting) object.”

The way I see it, in the case of the convergent sequence there are two mathematical objects – the resulting limit and the convergent sequence. In the case of the divergent sequence there is only one object – the divergent sequence.

I added the correction that follows two hours later after the original post: In the case of the divergent sequence there are also two objects – the divergent sequence and the infinity.

Re: “"Note that one can turn a process into an object by simply giving it a name." Yes, you are right, but only if one defined this process before and if this definition is free from contradiction.”

I do not think this is how mathematics works. There are no infinite straight lines and yet Euclidean geometry is internally consistent. Removing this axiom makes other geometries possible. Does this mean that Euclidean geometry is wrong? This example mirrors our problem with of "potential and actual" infinity.  

Hi Vadim, 

Re: "Forming the set of all natural numbers is not an act of completing an infinite process. (Marcel) ... Yes, of course. In this case we deal with potential infinity. ... (Vadim)"

I think you misunderstood Marcel's message which contradicts your interpretation. 

Re: "Re: "The most famous contradiction in set theory is the one of Russell… Note the absence of the word "infinite"."

I think that Russell paradox has nothing to do with infinity. It relates to the set theory in general and its axioms. This is why Marcel pointed out to the absence of the word "infinite".

Quoting Vadim

[...] that the number ω must be the first integer which comes behind all numbers v, i.e. which could be called larger than any number ν.

The set of all natural numbers is defined to be ω and apparently you have a problem using it before "finishing the naturals".

Are you allowed to use pi or square root of 2 only after listing its infinity of decimals? Or to use the number of one trillion without ever having counted that far? It is the definition of pi, of square root of 2, or of one trillion, that you need, not the listing. 

This is the great misunderstanding about the notion of set: associating it with a process of listing its members. That may be useful in simple situations. But sets are basically properties with a label of being "objectifyable" (which simply means that they can be a subject of statements). Unlike material collections (art, stamps,...) one does not need to "possess a set". I do not have the money nor the time to really collect all post stamps ever produced by any country. Just give me a good catalogue and I'll come a fair way answering reasonable questions about this set. In the same way, one can use logic and set axioms as a guide to solve reasonable problems involving infinite (or extremely large finite) sets without a process collecting its elements exhaustively.

Quoting Vadim:

The absence of the word "infinite" does not mean the absence of infinity. The postulated set in the paradox is infinite. Unfortunately, this set does not exist as well as the Set of all sets. Only the absurd idea of actual infinity lead to the contradictory inference of the existence of such sets.

Read my previous post carefully: The barber paradox (which uses the same logic formula) is about an imaginary human society and an attempt to define a barber within that society. This example is as finite as finite can be. The mere attempt of defining a mathematical object x by

(*)    (all y) ( not P(y,y) P(y,x) )

-- whatever statement is meant by P(.,.) -- is pointless as it delivers

        not P(x,x) P(x,x)       (generic y replaced by x).

You could as well attempt to define x to be a green square that is blue, given that green isn't blue. Would you call this description a paradox in "the theory of colored squares"?

Therefore, I explicitly confirm my earlier statements: the so-called Russell paradox finally has nothing to do with infinity. It is not a paradox either, it is just a clever contradictio in terminis. Cantor's original setup failed to prevent such pointless definitions.

RE:  I mean traditional mathematical sense: "to exist" means "to be free from contradiction" (Poincare).

Peano arithmetic has been shown by Godel to be incomplete or inconsistent. Even adding more axioms will not resolve the incompleteness. Anyway, Peano arithmetic is potentially contradictory. Is there anything of sufficient interest existing by Poincare's standards?

The pragmatic view of most pure mathematicians is (I think) that something exists mathematically if either it is required explicitly in some axiom, or its existence can be derived as a consequence of the axioms. If one day a contradiction would show up, we make a hopefully better proposal of axioms and reset our notion of existence to the new standard. This is basically what was done in the early 20th century after Cantor provided an insufficient axiomatic basis of ("naive") set theory and Russell produced his antinomy.

Our mathematics is a human affair after all.

My full and enthusiastic approval with Marcel's statement: "Our mathematics is a human affair after all." In a hopefully forthcoming book that I co-authored with Polish colleague Anna Rybak I put the same point in the following way: "Mathematics is a matter of mutual agreement."

Dear Mr. István Lénárt, I agree with Marcel and you that “Our science is a human affair after all".  The problem is: it is difficult to “settle down the agreement”. Because the “settle down work” may be the pioneering work and be against the current trend. We human have been trying very hard to “settle down the agreement for being free from contradictions” since Zeno’s time, but nothing has been done to solve the defects disclose by the infinite related paradoxes and the “contradictions” (they keep challenging our intelligence) in present classical science theory system for more than 2500 years-------it is very pity that we are still living in Zeno’s time.

Let’s word hard to settle down mutual agreement.

Best regards,

Geng

I think claiming that our current understanding of infinity is completely wrong is not an effective strategy for introducing new ideas. Surely, such claims get immediate attention, but they also antagonise readers and attract unprecedented scrutiny for which the author of claims has to be ready.

Right from his first post, I have been very sympathetic towards Geng’s desire to contribute to our understanding of infinity. Unfortunately, I cannot say that I understand what it is that he is proposing, and this is not for lack of trying on my part. Instead of convincing his readers that mathematicians have been wrong for thousand years, I would prefer him to provide a clear explanation of the proposed revision or addition to our knowledge.

It is not an easy process though. It requires patience and ingenuity because human communication is rarely direct. Not knowing the language should not be used as an excuse as there are many other ways of explaining such as formal definitions and axioms, images, diagrams, videos, etc. The responsibility to elucidate is on the person who makes claims, not the other way around. For various reasons, even with the best intention of the author there is no guarantee that his ideas will be accepted or even discussed. This is an unfortunate but true fact of social life dynamics.  

Dear Miguel, thank you for your interesting view point. I find the key work in your post: “misinterpretation” and the following 3 things are our major differences:

1, I don’t believe that it is scientifically that different people can have different understandings, different definitions and different interpretations for “infinite”-------there can be only one definition for “infinite” in our science.

2, many people and I believe that there is still the suspended Ancient Zeno’s Achilles--Turtle Race Paradox and it is impossible to be solved (defended) in our present classical science theory system (you can check on internet).

3, I believe that there is really “Harmonic Series Paradox of strict mathematical proven Ancient Zeno’s Achilles--Turtle Race Paradox” in our present classical science theory system-------- the suspended Ancient Zeno’s Achilles--Turtle Race Paradox have been keeping challenging us human for more than 2500 years. I know you don’t, but would you (or any other scientific researchers in the world) please so kind as to point out what exactly the mistakes in my post (being posted many times in different threads in RG) about that “Harmonic Series Paradox of strict mathematical proven Ancient Zeno’s Achilles--Turtle Race Paradox”?

Looking forward to you frank ideas.

Yours,

Geng

 Dear Mr. Untangling Mathl, thank you for your frank view point and I really admire and appreciate your clear and logical posts. All my works against the traditional theory of “infinite” in our present classical science theory system were aroused from that “Harmonic Series Paradox of strict mathematical proven Ancient Zeno’s Achilles--Turtle Race Paradox”.

As a re--cognizing fruit to certain knowledge system, paradox in our science system is a kind of amazing treasures closely related to the philosophy of science. In a special way, it purposely discloses certain defects in some of our scientific cognitive ideas and behaviors as well as some fundamental defects in our present science system. The exactly same producing mechanism and surviving conditions decide those defects and their relating paradoxes coexist with the same fate（birth or death）in our science system. So,（1）the paradoxes are unavoidable if the relating defects are not solved, not only those paradoxes are not be able to disappear themselves but their family members will increase more and more keeping disclosing the same defects from different angles;（2）the paradoxes relating defects are unavoidable, these fundamental defects will mislead our working ideas, result in mistaken cognitive behaviors and fruits in our science;（3）it is impossible to solve those paradoxes produced by the innate fundamental defects of the relating science system in the theoretical frame of this very same science system itself------ the newly born “Paradox of Harmonic Series” has been a typical example.

 Yours,

Geng

Dear Miguel, thank you for your feedback, it is good that we can have so frank exchanges in our work. I really need different view points. I am now presenting you my frank feedbacks.

I am very sorry to say that your (of cause, some other people’s too) definition on infinite is really un-scientific because you forget the whole half part of infinite--------the “infinitesimal” related field. So, I now know why you can not understand all the things happen in the field of “infinitesimal” related mathematics, it is difficult for you to understand what “Ancient Zeno’s Achilles--Turtle Race Paradox” is and what “Harmonic Series Paradox” is. Do you really think that “Ancient Zeno’s Achilles--Turtle Race Paradox” can be solved by your way of ”just draw the graphic of space run, and you will see that Achilles overtakes the turtle”.

During my 40 years work in the infinite related field of science, on the one hand I have been really trying very hard to do the “self--check” process and I am sure more than you just because what I am doing is against the current trend but you are not, on the other hand I am trying very hard purposely to invite people give their frank ideas pointing out what exactly my mistakes are so that I can think deeper and from different angles, or make me more sure about my work.

I hope you can forgive my frankness saying something you don’t like to hear.

My heartfelt thanks to you,

Geng

Hi Geng, 

Regarding: "[Cantor's?] definition on infinite is really un-scientific  because you forget the whole half part of infinite--------the “infinitesimal” related field."

It looks to me that you are attempting to unify the theory of infinity with theories of infinitesimals. I do not think this is possible mainly due to the way human mind seems to work which explains why contemporary mathematics is the way it is. Nevertheless, could you describe in an accessible way to us mortals how do you propose to achieve this? 

Dear Mr. Untangling Mathl and Mr. Miguel Ángel Montes, thank you so much for your questions! You are right that I attempt to unify the theory of infinity with theories of infinitesimals.

1, we have “big” and “small” in our science and the related numerical cognizing way to universal things around us. So, we have “very big (such as 10001000)” and “very small (such as 1/10001000)”, “extremely big” and “extremely small”… in mathematics. When “infinite” came into our science and mathematics, we naturally and logically have “infinite big-----infinities” and “infinite small-----infinitesimals”. According to my humble idea, to unify the theory of “infinite big-----infinities” with theories of “infinite small----- infinitesimals” within one scientific theory frame is a very important task for human scientific researchers; our scientific infinite theory system should cover both “infinite big-----infinities” and “infinite small-----infinitesimals”. This is the very reason to call for A Revolution in the Infinite Related Foundation of Mathematics.

2, when we study Cantor’s work, we can find that he is really a great mathematician who contributed greatly to some work in the numerical cognizing of “infinite big-----infinities” related applied mathematics. But he didn’t do anything on the portion of another half------ the field of “infinitesimal” related mathematics; that is why I call Cantor’s infinite theory “half infinite” in my papers. But this is not Cantor’s fault, it is because of the 2500 years “potential infinite—actual infinite” and the split of “potential infinite—actual infinite” theory system in our science.

3, my RG friend Miguel is right that my “claims, arguments, examples, have always been the same”------I am keeping one direction working so hard and suffering so much to dispel “the huge black cloud of infinite related paradoxes over mathematics sky”. My work on A Revolution in the Infinite Related Foundation of Mathematics is a pioneering one and I am so far only doing the beginning which is introduced in about 30 of my papers during past 25 years (8 of them have been uploaded on RG).

4, I have known from very beginning that my idea and my work is against our current trend, but the “huge black cloud” has been right there challenging us human for more than 2500 years and it is impossible to run away from it. 2500 years pass in a flap; I will try my best to do something and I am very sure the “huge black cloud” must be dispelled by someone as long as we human still alive and our science being developing.

Thank you again!

Yours,

Geng

Dear Mr. Ellis D. Cooper

Do you believe there is suspended “Ancient Zeno’s Achilles--Turtle Race Paradox” in (our) science?

Best regards, 

Geng

Dear Mr. Ellis D. Cooper, thank you.

Would you please check on the 5th page in this thread about the “Harmonic Series Paradox” I cited from my newly published paper?

Your frank ideas are always warmly welcomed and I sincerely wish you can point out the “mistakes” in it mathematically or philosophically frankly.

Yours sincerely,

Geng

Dear Mr. Ellis D. Cooper, thank you.

1, do you mean that you do not agree with the "brackets-placing rule" applied in the divergence proof of Harmonic Series in present mathematics theory system so the proof is wrong? Would you please tell why？

2, it is a very easy citation: just check current higher mathematical books written in all kinds of languages about the divergence proof of Harmonic Series.

Ok, I will improve that post and deled the history of it “given by Oresme in about 1360” in the post.

A very good discussion, thank you again!

Yours sincerely,

Geng

In mathematics one can rarely say that one topic is unrelated to another. With due care, I wish to state that infinity as a concept in set theory has little to do with infinitesimals and, in fact, it also has little to do with "infinite reals" as these are not cardinal numbers.

Infinite elements can only be described in the language of ordered fields. Such fields are always infinite sets, but many (like the standard reals) do not contain infinite elements. When an infinite element x occurs, 1/x is an infinitesimal. There is nothing paradoxical with them.

I also recall a fairly standard habit to add a formal "infinite element" ∞ to the real line, often specified with + or -, as a symbol to indicate a runaway process. Usually it goes with an agreement that 1/∞ = 0 to avoid introducing infinitesimals. It is nothing but a comfortable notation with no deep thoughts or philosophical considerations.

I cannot imagine anything Cantor could have done with infinitesimals or infinite (nonstandard) reals in general set theory. It  is simply a different topic. You are seeing ghosts, Geng.

Dear Colleagues,

Good Day,

“Two things are infinite: the universe and human stupidity; and I'm not sure about the universe.”

                                                                  ― Albert Einstein

Dear Colleagues,

Good Day,

Resources: Finite and Infinite Resources

Finite and Infinite

Resources are not found in Nature. They are created. It is the human mind that creates a resource out of natural objects or social circumstances when it discovers a value for them.

Thus, there is no inherent limit to the potential productivity of resources other than that which is imposed by the limits of our vision and imagination.

Please, see the link:

http://humanscience.wikia.com/wiki/Resources:_Finite_and_Infinite_Resources

Geng, also please stop connecting the harmonic series with Zeno's "paradox". The harmonic series with sum of 1/n for n >= 1 is well- known to diverge while any computation with reasonable assumptions about Achilles and the tortoise leads to a rapidly converging sequence.

Tha magic "bracket placing rule" in Oresme's proof  can be presented as a finite computation showing that the sum of 1/n for n running from 1 to 2k exceeds 1+k/2 (integer k). It is totally convincing, there are no mysteries or paradoxes involved.

My eldest granddaughter (18) recently went in discussion with me after her first course in philosophy, which dealt a.o. with Parmenides and Zeno. In defense of them (she called them "idiots"), I explained to her that they had a philosophical problem understanding continuous movement and change. Their role in early philosophy is their major merit, there is little of actual value in their thinking, most certainly not in mathematics or physics.

Dear Mr. Hazim Hashim Tahir, thank you for your insightful ideas. I agree with you that science is our human’s and the world can be scientific.

Best regards, 

Geng

Dear Mr. Marcel Van de Vel, thank you for your understandings and kind suggestions. You are right that I have discovered the suspended defects (I am seeing ghosts) in present classical science theory system and I will try my best and keep working hard to solve the defects.

Best regards, 

Geng

Thank you my dear Mr. downvoter, would you be so kind as to join the discussion and tell me your frank point of view what mistakes in the above question and how to correct the mistakes. That is what we really need in “question--discussion”-----not just hiding somewhere in the dark and killing. Well, it is ok different people have different enjoyments.

I have known from very beginning more than 40 years ago that my idea and my work is against our current trend and I have suffered a lot in my life already------I can stand more downvotes.

Although my RG score is getting lower and lower week by week by the downvoters, I still warmly welcome opposite points of view in my “question--discussion” from which I benefit a lot.

Beg you and thank you again for your kind participation.

Yours,

Geng

Dear Geng, I send you the greetings of a congenial spirit - mine. Definitions of infinity are rather out of my own scope of interest, because I smell philosophy just as much as math in this discussion. I have no sense for philosophy; my real interest lies in building bridges between concrete and abstract, practice and theory, reality and mathematics. However, I am deeply impressed by your efforts to adhere to your vision, to follow your inner eye. I think that this kind of stubbornness can fruit new results, give birth to new sciences. I wish you many successes in your further fights.

Dear Mr. István Lénárt, I am very grateful to your encouragement and spiritual support.

Our interests may meet someday because part of my work is to build bridges between concrete and abstract, practice and theory, reality and mathematics.

I am not sure how long I can stay in this world but I am very sure to try my best working hard in the new field in the rest of my life.

Thank you again.

Best regards, 

Geng

I think that the proper question would be: Are you able to see the difference between actual and potential infinity?

The great french mathematician Poincaré was able to see the difference and stated the following definition.

Actual infinity does not exist. What we call infinite is only the endless possibility of creating new objects no matter how many exist already.

Poincaré (1854-1912).

Our mind works handling abstractions. When we say that A and B are the same, simply, we neglect those properties that A satisfies and B does not. In some given context these properties can be neglected, but if you do not define the context, hardly your claims can be accurate.

For instance, in geometry, the points A=(0,1) and B=(1,1) are not the same because of its different location. By contrast, in physics, we can consider that B is the point A that has moved to the location (1,1).

According to the Pointcaré's definition, a potential infinite process cannot be ended, while an actual infinite one can be completed and ended.

For instance, consider the following induction process.

1) The number 1 is a finite integer.

2) If n is finite, its succesor (n+1) is finite too.

In the scope of potential infinity, by this induction process we deduce that all integers are finite. By contrast, under the assumption of actual infinity, this induction process fails, because the infinite can be obtained. As a consequence, there is a number n that either has no succesor or n is finite and n+1 infinite.

Can you see the difference?